Recently I came across some Economics notes which wrote the profit maximization equation as follows:
$\underset{p}\max \; \pi = p.q(p) - p.c(p)$
where, $p$ is the price, $q(p)$ is the quantity at that price and $c(p)$ is the cost at the given price.
I think that the $\max$ on the LHS is not meaningful as there is no $p$ there and it is more correct/meaningful to write the equation as follows.
$\pi = \underset{p}\max \; p.q(p) - p.c(p)$
Are there any conventions/guidelines for this? Are they both equivalent?
If you want to express the term on the rhs gets maximal if you plug in $p=π$, you can use argmax, like:
$$\pi = \underset{p}{\text{argmax}} ( pq(p) - pc(p)).$$
If you want to express that π is the maximum of the rhs write: $$\pi = \underset{p}{\text{max}} ( pq(p) - pc(p)).$$
The term $\underset{p}{\max} \pi$ is well-defined, too. And it is $$\underset{p}{\max} \pi = \pi.$$ Hence: $$ \underset{p}{\text{max}}\pi = pq(p) - pc(p) \qquad ⇔\qquad \pi = pq(p) - pc(p)$$
which is most likely not what you want to express.